On zeros of polynomials and allied functions satisfying second order differential equations

Differential, Difference and Asymptotic Relations for Pollaczek-Jacobi Type Orthogonal Polynomials and Their Hankel Determinants

In this paper, we study the orthogonal polynomials with respect to a singularly perturbed Pollaczek-Jacobi type weight $$ w(x,t):=(1-x^2)^α\mathrm{e}^{-\frac{t}{1-x^{2}}},\qquad x\in[-1,1],\;\;α>0,\;\;t>0. $$ By using the ladder operator approach, we establish the second-order difference equations satisfied by the recurrence coefficient $β_n(t)$ and the sub-leading coefficient $\mathrm{p}(n,t)$ of the monic orthogonal polynomials, respectively. We show that the logarithmic derivative of $β_n(t)$ can be expressed in terms of a particular Painlevé V transcendent. The large $n$ asymptotic expansi…

Orthogonal matrix polynomials satisfying differential equations with recurrence coefficients having non-scalar limits

We introduce a family of weight matrices $W$ of the form $T(t)T^*(t)$, $T(t)=e^{\mathscr{A}t}e^{\mathscr{D}t^2}$, where $\mathscr{A}$ is certain nilpotent matrix and $\mathscr{D}$ is a diagonal matrix with negative real entries. The weight matrices $W$ have arbitrary size $N\times N$ and depend on $N$ parameters. The orthogonal polynomials with respect to this family of weight matrices satisfy a second order differential equation with differential coefficients that are matrix polynomials $F_2$, $F_1$ and $F_0$ (independent of $n$) of degrees not bigger than 2, 1 and 0 respectively. For size $2…

满足奇数阶微分方程的正交矩阵多项式的一些例子

Some examples of orthogonal matrix polynomials satisfying odd order differential equations

本文给出不可约化为标量权重的权重矩阵实例，其对应的矩阵正交多项式序列成为奇数阶线性微分算子的本征函数，突破了标量情形下阶数必须为偶数的限制。

Analytical solutions of the Riccati equation with coefficients satisfying integral or differential conditions with arbitrary functions

Ten new exact solutions of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ are presented. The solutions are obtained by assuming certain relations among the coefficients $a(x)$, $b(x)$ and $c(x)$ of the Riccati equation, in the form of some integral or differential expressions, also involving some arbitrary functions. By appropriately choosing the form of the coefficients of the Riccati equation, with the help of the conditions imposed on the coefficients, we obtain ten new integrability cases for the Riccati equation. For each case the general solution of the Riccati equation is also presen…

Using $\D$-operators to construct orthogonal polynomials satisfying higher order difference or differential equations

We introduce the concept of $\D$-operators associated to a sequence of polynomials $(p_n)_n$ and an algebra $\A$ of operators acting in the linear space of polynomials. In this paper, we show that this concept is a powerful tool to generate families of orthogonal polynomials which are eigenfunctions of a higher order difference or differential operator. Indeed, given a classical discrete family $(p_n)_n$ of orthogonal polynomials (Charlier, Meixner, Krawtchouk or Hahn), we form a new sequence of polynomials $(q_n)_n$ by considering a linear combination of two consecutive $p_n$: $q_n=p_n+β_np_{…

On the convergence of formal Dulac series satisfying an algebraic ODE

We propose a sufficient condition of the convergence of a Dulac series formally satisfying an algebraic ordinary differential equation (ODE). Such formal solutions of algebraic ODEs appear rather often, in particular, the third, fifth, and sixth Painlevé equations possess formal Dulac series solutions, whose convergence follows from the proposed sufficient condition.

Asymptotic behaviors of linear advanced systems of differential equations

In this paper, we use a Banach fixed point theorem to obtain suficient conditions satisfying the convergence and exponential convergence of solutions for the linear system of advanced differential equations. The considered system with multiple variable advanced arguments is discussed as well. The obtained theorems generalize previous results of Dung [8], from the one dimension to the n dimension.

Liouvillian Solutions of First Order Non Linear Differential Equations

Let $k$ be a differential field of characteristic zero and $E$ be a liouvillian extension of $k$. For any differential subfield $K$ intermediate to $E$ and $k$, we prove that there is an element in the set $K-k$ satisfying a linear homogeneous differential equation over $k$. We apply our results to study liouvillian solutions of first order non linear differential equations and provide generalisations and new proofs for several results of M. Singer and M. Rosenlicht on this topic.

Majorana transformation for differential equations

We present a method for reducing the order of ordinary differential equations satisfying a given scaling relation (Majorana scale-invariant equations). We also develop a variant of this method, aimed to reduce the degree of non-linearity of the lower-order equation. Some applications of these methods are carried out and, in particular, we show that second-order Emden-Fowler equations can be transformed into first-order Abel equations. The work presented here is a generalization of a method used by Majorana in order to solve the Thomas-Fermi equation.

On Conjectures of A. Eremenko and A. Gabrielov

We study polynomials p(x) satisfying a differential equation of the form p"(x)-h'(x)p'(x)+H(x)p(x)=0, where h(x)=x^3/3+ax. We prove a conjecture of A. Eremenko and A. Gabrielov.

On the algebraic dependence of holonomic functions

We study the form of possible algebraic relations between functions satisfying linear differential equations. In particular , if f and g satisfy linear differential equations and are algebraically dependent, we give conditions on the differential Galois group associated to f guaranteeing that g is a polynomial in f. We apply this to hypergeometric functions and iterated integrals.

Krylov-Röckner条件下随机微分方程的逐路径唯一性

Path-by-path uniqueness for stochastic differential equations under Krylov-Röckner condition

本文证明了在Krylov-Röckner条件下，由布朗运动驱动的随机微分方程具有逐路径唯一性，从而得到强完备性并构成随机动力系统，并讨论了其在边值问题中的应用。

涉及旋度算子的微分包含

Differential Inclusions Involving the Curl Operator

该文研究涉及旋度算子的微分包含问题，探讨满足特定旋度条件的Sobolev空间函数的存在性，明确了定义域与集合的相关设定。

On a functional-differential equation with quasi-arithmetic mean value

In this paper we describe all differentiable functions $φ,ψ\colon E\to\mathbb{R}$ satisfying the functional-differential equation \begin{equation*} [φ(y) - φ(x)]ψ'\bigl(h(x,y)\bigr) = [ψ(y) - ψ(x)]φ'\bigl(h(x,y)\bigr), \end{equation*} for all $x,y\in E$, $x<y$, where $E \subseteq \mathbb{R}$ is a nonempty open interval, $h(\cdot,\cdot)$ is a quasi-arithmetic mean, i.e. $h(x,y)=H^{-1}(αH (x)+βH (y))$, $x,y\in E$, for some differentiable and strictly monotone function $H\colon E \to H(E)$ and fixed $α, β\in (0,1)$ with $α+β=1$.

Picard--Vessiot structures

We demonstrate existence and uniqueness of Picard--Vessiot extensions satisfying prescribed properties, for systems of linear differential equations over a field satisfying the same properties, under some closure assumptions on the field of constants. An example includes the case of a equations over a real field, with a real-closed field of constants. The result is obtained through a model theoretic approach.

Solvability of dissipative second order left-invariant differential operators on the Heisenberg group

We prove local solvability for large classes of operators of the form $$ L=\sum_{j,k=1}^{2n}a_{jk}V_jV_k+iαU,$$ where the $V_j$ are left-invariant vector fields on the Heisenberg group satisfying the commutation relations $[V_j,V_{j+n}]=U$ for $1\le j\le n$, and where $A=(a_{jk})$ is a complex symmetric matrix with semi-definite real part. Our results widely extend all of the results for the case of non-real, semi-definite matrices $A$ known to date, in particular those obtained recently jointly with F. Ricci under Sjöstrand's cone condition.

On Stability for Impulsive Delay Differential Equations and Application to a Periodic Lasota-Wazewska Model

We consider a class of scalar delay differential equations with impulses and satisfying an Yorke-type condition, for which some criteria for the global stability of the zero solution are established. Here, the usual requirements about the impulses are relaxed. The results can be applied to study the stability of other solutions, such as periodic solutions. As an illustration, a very general periodic Lasota-Wazewska model with impulses and multiple time-dependent delays is addressed, and the global attractivity of its positive periodic solution analysed. Our results are discussed within the con…

Fourier decay bound and differential images of self-similar measures

In this note, we investigate $C^2$ differential images of the homogeneous self-similar measure associated with an IFS $\mathcal{I}=\{ρx+a_j\}_{j=1}^m$ satisfying the strong separation condition and a positive probability vector $\vec{p}$. It is shown that the Fourier transforms of such image measures have power decay for any contractive ratio $ρ\in (0, 1/m)$, any translation vector $\vec{a}=(a_1, \ldots, a_m)$ and probability vector $\vec{p}$, which extends a result of Kaufman on Bernoulli convolutions. Our proof relies on a key combinatorial lemma originated from Erdős, which is important in …

Ordinary differential equations with point interactions: An inverse problem

Given a linear ordinary differential equation (ODE) on $\RE$ and a set of interface conditions at a finite set of points $I \subset \RE$, we consider the problem of determining another differential equation whose {\it global} solutions satisfy the original ODE on $\RE \backslash I $, and the interface conditions at $I $. Using an extension of the product of distributions with non-intersecting singular supports presented in [L. Hörmander, The Analysis of Linear Partial Diffe\-rential Operators I, Springer-Verlag, 1983], we determine an {\it intrinsic} solution of this problem, i.e. a new ODE, s…